Efficient Algorithms for the Order Preserving Pattern Matching Problem

0.52 MB·English

by  Simone Faro

Efficient Algorithms for the
Order Preserving Pattern Matching Problem(cid:63)
Simone Faro† and M. O˘guzhan Ku¨lekci‡
†Universita` di Catania, Department of Mathematics and Computer Science, Italy
‡Istanbul Medipol University, Department of Biomedical Engineering, Turkey
5 [email protected], [email protected]
1
0
2
Abstract. Givenapatternxoflengthmandatextyoflengthn,both

n overanorderedalphabet,theorder-preservingpatternmatching problem
a consistsinfindingallsubstringsofthetextwiththesamerelativeorder
J as the pattern. It is an approximate variant of the well known exact

6 pattern matching problem which has gained attention in recent years.
1 Thisinterestingproblemfindsapplicationsinalotoffieldsastimeseries

analysis, like share prices on stock markets, weather data analysis or to
]
S musical melody matching. In this paper we present two new filtering
D approaches which turn out to be much more effective in practice than
thepreviouslypresentedmethods.Fromourexperimentalresultsitturns
.
s outthatourproposedsolutionsareupto2timesfasterthantheprevious
c solutions reducing the number of false positives up to 99%.
[

1
1 Introduction
v
1
0 Given a pattern x of length m and a text y of length n, both over a common
0 alphabet Σ, the exact string matching problem consists in finding all occur-
4
rences of the string x in y. String matching is a very important subject in the
0
. wider domain of text processing and algorithms for the problem are also basic
1
components used in the implementations of practical softwares existing under
0
most operating systems. Moreover, they emphasize programming methods that
5
1 serve as paradigms in other fields of computer science. Finally they also play an
: important role in theoretical computer science by providing challenging prob-
v
lems. The worst case lower bound of the string matching problem is (n) and
i
X was achieved the first time by the well known algorithm by Knuth, MOorris and
r Pratt[8].Howevermanystringmatchingalgorithmshavebeenalsodevelopedto
a
obtainsublinear (nlogm/m)performanceonaverage.AmongthemtheBoyer-
O
Moore algorithm [1] deserves a special mention, since it has been particularly
successful and has inspired much work.
The order-preserving pattern matching problem [2, 3, 8, 9] (OPPM in short)
is an approximate variant of the exact pattern matching problem which has
(cid:63) This work has been supported by the Scientific & Technological Research Council
OfTurkey(TUBITAK),theDepartmentOfScienceFellowships&GrantPrograms
(BIDEB), 2221 Fellowship Program, and by G.N.C.S., Istituto Nazionale di Alta
Matematica “Francesco Severi”.
y = 8 11 10 16 15 20 13 17 14 18 20 18 25 17 24 25 26
x = 6 5 8 4 7
Fig.1. Example of a pattern x of length 5 over an integer alphabet with two order
preserving occurrences in a text y of length 17, at positions 3 and 10.
gained attention in recent years. In this variant the characters of x and y are
drawnfromanorderedalphabetΣ withatotalorderrelationdefinedonit.The
task of the problem is to find all substrings of the text with the same relative
order as the pattern.
Forinstancetherelativeorderofthesequencex= 6,5,8,4,7 isthesequence
(cid:104) (cid:105)
3,1,0,4,2 since6hasrank3,5asrank1,andsoon.Thusxoccursinthestring
(cid:104) (cid:105)
y = 8,11,10,16,15,20,13,17,14,18,20,18,25,17,20,25,26 atposition3,since
(cid:104) (cid:105)
xandthesubsequence 16,15,20,13,17 sharethesamerelativeorder.Another
(cid:104) (cid:105)
occurrence of x in y is at position 10 (see Fig.1).
The OPPM problem finds applications in the fields where we are interested
in finding patterns affected by relative orders, not by their absolute values. For
example, it can be applied to time series analysis like share prices on stock
markets, weather data or to musical melody matching of two musical scores.
In the last few years some solutions have been proposed for the order-
preserving pattern matching problem. The first solution was presented by Ku-
bica et al. [7] in 2013. They proposed a (n+mlogm) solution over generic
O
orderedalphabetsbasedontheKnuth-Morris-Prattalgorithm[8]anda (n+m)
O
solution in the case of integer alphabets. Some months later Kim et al. [6] pre-
sented a similar solution running in (n+mlogm) time based on the KMP
O
approach. Although Kim et al. stressed some doubts about the applicability of
the Boyer-Moore approach [1] to order-preserving matching problem, in 2013
Cho et al. [3] presented a method for deciding the order-isomorphism between
two sequences showing that the Boyer-Moore approach can be applied also to
the order-preserving variant of the pattern matching problem. More recently
Chhabra and Tarhio [2] presented a more practical solution based on approx-
imate string matching. Their technique is based on a conversion of the input
sequences in binary sequences and on the application of any standard algorithm
for exact string matching as a filtration method.
In this paper we present two new families of filtering approaches which turn
outtobemuchmoreeffectiveinpracticethanthepreviouslypresentedmethods.
WhilethetechniqueproposedbyChhabraandTarhiotranslatestheinputstrings
in binary sequences, our methods work on sequences over larger alphabets in
ordertospeedupthesearchingprocessandreducethenumberoffalsepositives.
Fromourexperimentalresultsitturnsoutthatourproposedsolutionsareupto
2 times faster than the previous solutions reducing the number of false positives
up to 99% under suitable conditions.
The paper is organized as follows. In Section 2 we give preliminary notions
and definitions relative to the order-preserving pattern matching problem while
in Section 3 we briefly describe the previous solutions to the problem. Then we
present our new solutions in Section 4 and evaluate their performances against
the previous algorithms in Section 5. Conclusions are drawn in Section 6.
2 Notions and Basic Definitions
A string x over an ordered alphabet Σ, of size σ, is defined as a sequence of
elements in Σ. We suppose that a total order relation “ ” is defined on it, so
≤
that we could establish if a b for each a,b Σ.
≤ ∈
We indicate the length of a string x with the symbol x. We refer to the
| |
elements in x with the symbol x[i], for 0 i < x. Moreover we indicate with
≤ | |
x[i...j] the subsequence of x from the element of position i to the element of
position j (including the extremes), for 0 i j < x.
≤ ≤ | |
Wesaythattwosequencesx,y Σ∗areorderisomorphiciftherelativeorder
∈
of their elements is the same. More formally we give the following definition.
Definition 1 (order isomorphism). Given an ordered alphabet Σ and two
sequencesx,y Σ∗ ofthesamelength,wesaythatxandyareorder-isomorphic,
∈
and write x y, if the following conditions hold
≈
1. x = y
| | | |
2. x[i] x[j] if and only if y[i] y[j], for 0 i,j < x
≤ ≤ ≤ | |
Definition 2 (rank function). Let x be a sequence of length m over an or-
dered alphabet Σ. The rank function of x if a mapping r : 0,1,...,m 1
{ − } →
0,1,...,m 1 such that x[r(i)] x[r(j)] holds for each pair 0 i < j < m.
{ − } ≤ ≤
Formally we define
r(i)= j : x[j]<x[i] or (x[j]=x[i] and j <i)
|{ }|
for 0 i<m.
≤
We will refer to the value r(i) as the rank of x[i] in x, while we will refer to the
sequence r(0),r(1),...r(m 1) as the relative order of x.
(cid:104) − (cid:105)
According to Definition 2 we have that x[r(0)] is the smallest number while
x[r(m 1)] is the greater number in x. If we assume that sort(x) is the time
−
required to sort all the elements of x, then it is easy to observe that the relative
order of x can be computed in (sort(x)) time.
O
In addition, we define the equality function of x which indicates which ele-
ments of the sequence are equal (if any). More formally we have the following
definition.
Noder-Isomorphism(r,eq,y,i)
1. for i←0 to |x|−1 do
2. if (y[r(i)]>y[r(j+i+1)]) then return false
3. if (y[r(i)]<y[r(j+i+1)] and eq(i)=1) then return false
4. if (y[r(i)]=y[r(j+i+1)] and eq(i)=0) then return false
5. return true
Fig.2. The function used to verify if two sequences x and y[i...i+|x|−1] are order
isomorphic. We assume that the function receives as input the parameter r and eq
which represent the rank function and the equality function of x, respectively.
Definition 3 (equality function). Let x be a sequence of length m over an
ordered alphabet Σ and let r be the rank function of x. The equality function of
x if a mapping eq : 0,1,...,m 2 0,1 such that, for each 0 i<m
{ − }→{ } ≤
(cid:26)
1 if x[r(i)]=x[r(i+1)]
eq(i)=
0 otherwise
Let r be the rank function of a string x, such that m = x, and let q be its
| |
equality function. It is easy to prove that x and y are order isomorphic if and
only if they share the same rank and equality function, i.e. if and only if the
following two conditions hold
1. y[r(i)] y[r(i+1)], for 0 i<m 1
≤ ≤ −
2. y[r(i)]=y[r(i+1)] if and only if q(i)=1, for 0 i<m 1
≤ −
Example 1. Let x = 6,3,8,3,10,7,10 and y = 2,1,4,1,5,3,5 two sequences
(cid:104) (cid:105) (cid:104) (cid:105)
ofsize7.Wehavethattherelativeorderofxis(1,3,0,5,2,4,6)whileitsequal-
ity function is eq(x[i]) = (1,0,0,0,0,1). The two string are order isomorphic
according to the definition given above, i.e. x y.
≈
The procedure to verify that two numeric sequences, x and y, are order
isomorphicisshowninFig.2.Itreceivesasinputthefunctionsrandq,computed
on x and returns a boolean value indicating if x y. The algorithm requires
≈
(m) time, where m is the length of the sequences. A mismatch occurs when
O
one of the three conditions of lines 2, 3 and 4, holds.
TheOPPMproblemconsistsinfindingallsubstringofthetextwiththesame
relativeorderasthepattern.Specificallywehavethefollowingformaldefinition.
Definition 4 (order preserving pattern matching). Let x and y be two
sequences of length m and n, respectively (and n > m), both over an ordered
alphabet Σ. The order preserving pattern matching problem consists in finding
all indexes i, with 0 i<n m, such that y[i...i+m 1] x.
≤ − − ≈
Ifanoccurrenceofthepatternxstartsatportioniofthetexty,wesaythat
x has an order-preserving occurrence at position i.
3 Previous Results
TheOPPMproblemhasdrawnparticularattentioninthelastfewyears,during
which some efficient results have been proposed.
The first algorithm to solve the OPPM problem was presented by Kubica et
al. in [7]. Their solution was an adaptation of the well Known Knuth-Morris-
Prattalgorithmfortheexactstringmatchingproblem,wherethefailfunctionis
adapted to compute the order-borders table. The authors proved that this table
can be computed in linear time in the length of the pattern x, if the relative
order of x is known in advance. The overall time complexity of the algorithm is
(n+mlogm), where m is the length of the pattern while n is the length of
O
the text. However in [3] Cho et al. proved that the algorithm presented in [7]
can decide incorrectly when there are equal values in the string.
The second algorithm based on Knuth-Morris-Pratt was presented later by
Kim et al. [6]. Their algorithm is based on the prefix representation and it is
further optimized according to the nearest neighbor representation. The prefix
representationisbasedonfindingtherankofeachintegerintheprefix.Itcanbe
computed easily by inserting each character to the dynamic order statistic tree
andthencomputingtherankofeachcharacterintheprefix.Thetimecomplexity
of computing such prefix representation is O(mlogm). The failure function is
thencomputedasintheKnuth-Morris-PrattalgorithminO(mlogm)time.The
overalltimecomplexityofthisalgorithmisO(n+mlogm).Again,thissolution
does not work properly when there are equal values in the pattern.
The first sublinear solution for the OPPM problem was presented by Cho et
al. in [3]. Their algorithm is an adaptation to OPPM of the well known Boyer-
Moore approach. They apply a q-grams technique, i.e. groups of q consecutive
charactersaretreatedasasinglecondensedcharacter,inordertomaketheshifts
longer. In this way, a large amount of text can be skipped for long patterns.
More recently Chhabra and Tarhio presented a new practical solution [2]
based on a filtration technique. Their algorithm translates the input sequences
in two binary sequences and then use any standard exact pattern matching
algorithm as a filtration procedure. In particular in their approach a sequence s
is translated in a binary sequence β of length s 1 according to the following
| |−
position
(cid:26)
1 if s[i] s[i+1]
β[i]= ≥ (1)
0 otherwise
foreach0 i< s 1.Thistranslationisuniqueforagivensequencesandcan
≤ | |−
beperformedonlineonthetext,requiringconstanttimeforeachtextcharacter.
Thuswhenacandidateoccurrenceisfoundduringthefiltrationphaseanad-
ditionalverificationprocedureisruninordertocheckfortheorder-isomorphism
of the candidate substring and the pattern. Despite its quadratic time complex-
ity, this approach turns out to be simpler and more effective in practice than
earlier solutions. It is important to notice that any algorithm for exact string
matchingcanbeusedasafiltrationmethod.Theauthorsalsoprovedthatifthe
underlyingfiltrationalgorithmissublinearandthetextistranslatedonline,the
overallcomplexityofthealgorithmissublinearonaverage.Experimentalresults
conducted in [2] show that the filter approach was considerably faster than the
algorithm by Cho et al.
ForthesakeofcompletenesswenoticethatCrochemoreetal.presentedin[4]
a solution for the offline version of the OPPM problem based on a new data
structure called order-preserving suffix tree. Their solution finds all occurrences
ofxiny in ((mlogn)/loglogm+z)wherez isthenumberofoccurrencesofx
O
in y. In this paper we concentrate on the online version of the OPPM problem.
4 New Efficient Filter Based Algorithms
In this section we present two new general approaches for the OPPM problem.
Bothofthemarebasedonafiltrationtechnique,asin[2],butweuseinformation
extracted from groups of integers in the input string, as in [3], in order to make
thefiltrationphasemoreeffectiveintermsofefficiencyandaccuracy,asdiscussed
below.
Textfiltration isalargelyusedtechniqueinthefieldofexactandapproximate
stringmatching.Specifically,insteadofcheckingateachpositionofthetextifthe
patternoccurs,itseemstobemoreefficienttofiltertextpositionsandcheckonly
when a substring looks like the pattern. When a resemblance has been detected
a naive check of the occurrence is performed. In literature filtration techniques
are generally improved by using q-grams, i.e. groups of adjacent characters of
the string which are considered as a single character of a condensed alphabet.
It is always convenient to use a filtration method which better and faster lo-
calizecandidateoccurrences,whichimplyaccuracyandefficiencyofthemethod,
respectively.
The accuracy of a filtration method is a value indicating how many false
positives are detected during the filtration phase, i.e. the number of candidate
occurrences detected by the filtration algorithm which are not real occurrences
of the pattern. The efficiency is instead related with the time complexity of the
procedureweuseformanagingq-gramsandwiththetimeefficiencyoftheoverall
searching algorithm. It is clear that these two values are strongly related since
a low accuracy implies an high number of false positives and, as a consequence,
a decrease in the performance of the searching algorithm.
When using q-grams, a great accuracy translates in involving greater values
of q. However, in this context, the value of q represents a trade-off between the
computational time required for computing the q-grams for each window of the
text and the computational time needed for checking false positive candidate
occurrences. The larger is the value of q, the more time is needed to compute
each q-gram. On the other hand, the larger is the value of q, the smaller is the
numberoffalsepositivesthealgorithmfindsalongthetextduringthefiltration.
In our approaches we make use of the following definition of q-neighborhood
of an element in an integer string.
Definition 5 (q-neighborhood). Given a string x of length m, we define the
q-neighborhood of the element x[i], with 0 i<m q, as the sequence of q+1
≤ −
elements from position i to i+q in x, i.e.the sequence x[i],x[i+1],...,x[i+q] .
(cid:104) (cid:105)
Both the filtration methods presented below translate the input sequence
in a target numeric sequence which is used for the filtration. Specifically each
positioni ofthe sequenceis associated witha numericvalue computedfrom the
structure of the q-neighborhood of the element x[i].
4.1 The Neighborhood Ranking Approach
Givenastringxoflengthm,wecancomputetherelativepositionoftheelement
x[i] compared with the element x[j] by querying the inequality x[i] x[j]. For
≥
brevity we will write in symbol β (i,j) to indicate the boolean value resulting
x
fromtheaboveinequality,extendingtheformaldefinitiongiveninEquation(1).
Formally we have
(cid:26)
1 if x[i] x[j]
βx(i,j)= 0 otherw≥ise (2)
Itiseasytoobservethatifβ (i,j)=1wehavethatr(i) r(j)(x[j]precedes
x
≥
x[i] in the ordering of the elements of x), otherwise r(i)<r(j).
The neighborhood ranking (nr) approach associates each position i of the
string x (where 0 i < m q) with the sequence of the relative positions
≤ −
betweenx[i]andx[i+j],forj =1,...,q.Inotherwordswecomputethebinary
sequence β (i,i+1),β (i,i+2),...,β (i,i+q) oflengthqindicatingtherelative
x x x
(cid:104) (cid:105)
positions of the element x[i] compared with other values in its q-neighborhood.
Ofcourse,wedonotincludeinthesequencetherelativepositionofβ(i,i),since
it doesn’t give any additional information.
Since there are 2q possible configurations of a binary sequence of length q
the string x is converted in a sequence χq of length m q, where each element
x −
χq[i], for 0 i<m q, is a value such that 0 χq[i]<2q.
x ≤ − ≤ x
More formally we have the following definition
Definition 6 (q-NR sequence). Given a string x of length m and an integer
q < m, the q-nr sequence associated with x is a numeric sequence χq of length
x
m q over the alphabet 0,...,2q where
− { }
q
χq[i]=(cid:88)(cid:0)β (i,i+j) 2q−j(cid:1), for all 0 i<m q
x x × ≤ −
j=1
Example 2. Letx= 5,6,3,8,10,7,1,9,10,8 beasequenceoflength10.The4-
(cid:104) (cid:105)
neighborhood of the element x[2] is the subsequence 3,8,10,7,1 . Observe that
(cid:104) (cid:105)
x[2]isgreaterthanx[6]andlessthanallothervaluesinits4-neighborhood.Thus
therankingsequenceassociatedwiththeelementofposition2is 0,0,0,1 which
(cid:104) (cid:105)
translates in a nr value equal to 1. In a similar way we can observe that the nr
sequence associated with the element of position 3 is 0,1,1,0 which translates
(cid:104) (cid:105)
in a nr value equal to 6. The whole 4-nr sequence of length 6 associated to x is
χ4 = 4,8,1,6,15,8 .
x (cid:104) (cid:105)
Neighborhood Ranking Example nr seq. �3[i]
x
x[i] x[i+1],x[i+2],x[i+3] 0,0,0 0
 h i
x[i+3] x[i] x[i+1],x[i+2] 0,0,1 1
  h i
x[i+2] x[i] x[i+1],x[i+3] 0,1,0 2
  h i
x[i+2],x[i+3] x[i] x[i+1] 0,1,1 3
  h i
x[i+1] x[i] x[i+2],x[i+3] 1,0,0 4
  h i
x[i+1],x[i+3] x[i] x[i+2] 1,0,1 5
  h i
x[i+1],x[i+2] x[i] x[i+3] 1,1,0 6
  h i
x[i+1],x[i+2],x[i+3] x[i] 1,1,1 7
 h i
Fig.3.The23possible3-neighborhoodrankingsequencesassociatedwithelementx[i],
andtheircorrespondingnrvalue.Intheleftmostcolumnweshowtherankingposition
ofx[i]comparedwithotherelementsinitsneighborhood(cid:104)x[i],x[i+1],x[i+2],x[i+3](cid:105).
The following Lemma 1 and Corollary 1 prove that the nr approach can be
used to filter a text y in order to search for all order preserving occurrences of a
pattern x. In other words it proves that
i x y[i...i+m 1] i χq =χq[i...i+m k] .
{ | ≈ − }⊆{ | x y − }
Lemma 1. Let x and y be two sequences of length m and let χq and χq the
x y
q-ranking sequences associated to x and y, respectively. If x y then χq =χq.
≈ x y
Proof. Let r be the rank function associated to x and suppose by hypothesis
that x y. Then the following statements hold
≈
1. by Definition 2 we have x[r(i)] x[r(i+1)], for 0 i<m 1;
≤ ≤ −
2. by hyphotesis and Def.1, y[r(i)] y[r(i+1)], for 0 i<m 1;
≤ ≤ −
3. then by 1 and 2, x[i] x[j] iff y[i] y[j], for 0 i,j <m 1;
≤ ≤ ≤ −
4. the previous statement implies that x[i] x[i+j] iff y[i] y[i+j]
≥ ≥
for 0 i<m q and 1 j <q;
≤ − ≤
5. by statement 4 we have that β (i,i+j)=β (i,j+j)
x y
for 0 i<m q and 1 j <q;
≤ − ≤
6. finally, by 5 and Definition 6, we have χq[i]=χq[i], for 0 i<m q.
x y ≤ −
This last statement proves the thesis. (cid:4)
Thefollowingcorollarypricesthatthenrapproachcanbeusedasafiltering.
It trivially follows from Lemma 1.
Compute-NR-Value(x,i,q)
1. δ←0
2. for j ←1 to q do
3. δ=(δ(cid:28)1)+β (i,i+j)
x
4. return δ
Fig.4.Thefunctionwhichcomputestheq-neighborhoodrankingvalueoftheelement
of position i in a sequence x. The value id computed in O(q) time.
Corollary 1. Let x and y be two sequences of length m and n, respectively.
Let χq and χq the q-ranking sequences associated to x and y, respectively. If
x y
x y[j...j+m 1] then χq[i]=χq[j+i], for 0 i<m q. (cid:4)
≈ − x y ≤ −
Fig. 4 shows the procedure used for computing the nr value associated with
the element of the string x at position i. The time complexity of the procedure
is (q). Thus, given a pattern x of length m, a text y of length n and an
O
integer value q < m, we can solve the OPPM problem by searching χq for all
y
occurrences of χq, using any algorithm for the exact string matching problem.
x
During the preprocessing phase we compute the sequence χq and the functions
x
r and q . When an occurrence of χq is found at position i the verification
x x x
procedure Noder-Isomorphism(r,q,y,i) (shown in Fig.2) is run in order to
check if x y[i...i+m 1].
≈ −
Since in the worst case the algorithm finds a candidate occurrence at each
text position and each verification costs (m), the worst case time complexity
O
of the algorithm is (nm), while the filtration phase can be performed with a
O
(nq)worstcasetimecomplexity.However,followingthesameanalysisof[2],we
O
easilyprovethatverificationtimeapproacheszerowhenthelengthofthepattern
grows, so that the filtration time dominates. Thus if the filtration algorithm is
sublinear, the total algorithm is sublinear.
4.2 The Neighborhood Ordering Approach
Theneighborhoodrankingapproachdescribedintheprevioussectiongivespar-
tialinformationabouttherelativeorderingoftheelementsintheq-neighborhood
of an element in x. The q binary sequence used to represent each element x[i] is
not enough to describe the full ordering information of a set of q+1 elements.
The q-neighborhood ordering (no) approach, which we describe in this sec-
tion, associates each element of the x with a binary sequence which completely
describes the ordering disposition of the elements in the q-neighborhood of x[i].
The number of comparisons we need to order a sequence of q +1 elements is
between q (the best case) and q(q+1)/2 (the worst case). In this latter case it
is enough to compare the element x[j], where i j < i+q, with each element
≤
x[h], where j <h i+q.
≤
Thuseachelementofpositioniinx,with0 i<m q,isassociatedwitha
≤ −
binarysequenceoflengthq(q+1)/2whichcompletelydescribestherelativeorder
Neighborhood Ordering Example NO seq. '4[i]
x
x[i],x[i+1],x[i+2] 0,0,0 0
h i h i
x[i],x[i+2],x[i+1] 0,0,1 1
h i h i
x[i+2],x[i],x[i+1] 0,1,1 3
h i h i
x[i+1],x[i],x[i+2] 1,0,0 4
h i h i
x[i+1],x[i+2],x[i] 1,1,0 6
h i h i
x[i+2],x[i+1],x[i] 1,1,1 7
h i h i
Fig.5. The 3! possible ordering of the sequence (cid:104)x[i],x[i+1],x[i+2](cid:105) and the corre-
sponding binary sequence (cid:104)β (i,i+1),β (i,i+2),β (i+1,i+2)(cid:105).
x x x
Compute-NO-Value(x,i,q)
1. δ←0
2. for k←q downto 1 do
3. for j ←1 to k do
4. δ=(δ(cid:28)1)+β (i+q−k,i+q−k+j)
x
5. return δ
Fig.6.Thefunctionwhichcomputestheq-neighborhoodrankingvalueoftheelement
of position i in a sequence x. The value is computed in O(q2) time.
of the susequence x[i,...,i+q]. Since there are (q+1)! possible permutations
of a set of q+1 elements, the string x is converted in a sequence ϕq of length
x
m q, where each element ϕq[i] is a value such that 0 ϕq[i]<q(q+1)/2.
− x ≤ x
More formally we have the following definition
Definition 7 (q-NO sequence). Given a string x of length m and an integer
q <m, the q-no sequence associated with x is a numeric sequence ϕq of length
x
m q over the alphabet 0,...,q(q+1)/2 where
− { }
q
(cid:88)(cid:16) (cid:17)
ϕq[i]= χk[i+q k] 2(k)(k−1)/2 , for all 0 i<m q (3)
x x − × ≤ −
k=1
Thus the q-no value associated to x[i] is the combination of q different nr
sequences χq[i], χq−1[i+1], ..., χ1[i+q 1].
x x x −
For instance the 4-no value associated to x[i] is computed as
ϕ4[i]=χ4[i] 26+χ3[i+1] 22+χ2[i+2] 2+χ1[i+3]
x x × x × x × x